Arturo Magidin
This note collects several results on the capability of $p$-groups of class two and prime exponent. Among the new results, we settle the 4-generator case for this class.
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We consider the capability of $p$ groups of class two and odd prime exponent. We use linear algebra and counting arguments to establish a number of new results. In particular, we settle the 4-generator case, and prove a sufficient condition based on the ranks of $G/Z(G)$ and $[G,G]$.
We show that if $G$ is any $p$-group of class at most two and exponent $p$, then there exist groups $G_1$ and $G_2$ of class two and exponent $p$ that contain $G$, neither of which can be expressed as a central product, and with $G_1$ capable and $G_2$ not capable. We provide upper bounds for ${\rm rank}(G_i^{\rm ab})$ in terms of ${\rm rank}(G^{\rm ab})$ in each case.
A group is called capable if it is a central factor group. We consider the capability of finite groups of class two and exponent $p$, $p$ an odd prime. We restate the problem of capability as a problem about linear transformations, which may be checked explicitly for any specific instance of the problem. We use this restatement to derive some known results, and prove new ones. Among them, we reduce the general problem to an oft-considered special case, and prove that a 3-generated group of class 2 and exponent $p$ is either cyclic or capable.
We consider the capability of $p$-groups of class two and odd prime exponent. The question of capability is shown to be equivalent to a statement about vector spaces and linear transformations, and using the equivalence we give proofs of some old results and several new ones. In particular, we establish a number of new necessary and new sufficient conditions for capability, including a sufficient condition based only on the ranks of $G/Z(G)$ and $[G,G]$. Finally, we characterise the capable groups among the 5-generated groups in this class.
We give a complete list of all the 70 class two groups of exponent p (p>2) and order p^k for k<9. For each of these groups the number of conjugacy classes is a polynomial in p, and the order of the automorphism group is a polynomial in p. In contrast, Marcus du Sautoy and Michael Vaughan-Lee have given an example of a class two group of exponent p and order p^9 for which neither the number of conjugacy classes, nor the order of the automorphism group is polynomial on residue classes (PORC).
A group is called capable if it is a central factor group. For each prime $p$ and positive integer $c$, we prove the existence of a capable $p$-group of class $c$ minimally generated by an element of order $p$ and an element of order $p^{1+\lfloor\frac{c-1}{p-1}\rfloor}$. This is best possible.
A note on Frobenius quotient for prime-power divisor of the exponent of finite groups
本文研究有限群指数素幂因子对应的Frobenius商，并给出最大Frobenius商不超过群阶最小素因子的有限群的完全分类。
Let $p$ be a prime number and $\mathbb{Z}_p=\mathbb{Z}/p\mathbb{Z}$. We study finite groups with abelian derived subgroup and exponent $p$ in terms of group extension data and their matrix presentations. We show a one-to-one correspondence between the following two sets: (i) the isoclasses of class 2 groups of exponent $p$ and order $p^{m+n}$ and with derived subgroup $\mathbb{Z}_p^n$, and (ii) the set $\text{Gr}(n,\text{AS}_m(\mathbb{Z}_p))/\text{GL}_m(\mathbb{Z}_p)$ of orbits of $\text{Gr}(n,\text{AS}_m(\mathbb{Z}_p))$ under the congruence action by $\text{GL}_m(\mathbb{Z}_p)$, where $\text{…
An equation over a group with one unknown is called regular if the exponent sum of the unknown is nonzero. In this paper we prove that some regular equations of exponent $rp^s$, where $r \in \mathbb{Z}$, $s \in \mathbb{N}$, $\gcd(r,p)=1$, over the group UT$_n(\mathbb{F}_p)$ ($n \geq 2$) are solvable in an overgroup isomorphic to UT$_{(n-1)p^s + 1}(\mathbb{F}_p)$. Applying this for $n=3$ we prove that any regular equation of exponent $rp^s$ over the Heisenberg $p$-group UT$_3(\mathbb{F}_p)$ is solvable in an overgroup isomorphic to UT$_{2p^s + 1}(\mathbb{F}_p)$. The proofs of these results are …
Although any finite Bol loop of odd prime exponent is solvable, we show there exist such Bol loops with trivial center. We also construct finitely generated, infinite, simple Bruck loops of odd prime exponent for sufficiently large primes. This shows that the Burnside problem for Bruck loops has a negative answer.
A group is said to be capable if it is the central factor of some group. In this paper, among other results we have characterized capable groups of order $p^2q$, for any distinct primes $p, q$, which extends Theorem 1.2 of S. Rashid, N. H. Sarmin, A. Erfanian, and N. M. Mohd Ali, {\em On the non abelian tensor square and capability of groups of order $p^2q$}, Arch. Math., \textbf{97} (2011), 299--306. We have also computed the number of distinct element centralizers of a group (finite or infinite) with central factor of order $p^3$, which extends Proposition 2.10 of S. M. Jafarian Amiri, H. Ma…
Modular Functions and the Monstrous Exponents
本文通过模函数方法解释怪物有限简单群阶的素数分解中的指数问题。
We study the free metabelian group $M(2,n)$ of prime power exponent $n$ on two generators by means of invariants $M(2,n)'\to \mathbb{Z}_n$ that we construct from colorings of the squares in the integer grid $\mathbb{R} \times \mathbb{Z} \cup \mathbb{Z} \times \mathbb{R}$. In particular we improve bounds found by M.F. Newman for the order of $M(2,2^k)$. We study identities in $M(2,n)$, which give information about identities in the Burnside group $B(2,n)$ and the restricted Burnside group $R(2,n)$.
Recognition by the set of exponents in the prime factorization of the product of element orders
本文利用元素阶乘积的素因子分解指数集合，证明了若干射影特殊线性群及循环群可由该指数集合（或结合群阶）唯一确定。
On exponentiation, $p$-automata and HNN extensions of free abelian groups
本文证明对任意素数p，自由阿贝尔群的广泛HNN扩张类都能由有限p-自动机忠实表示，揭示了群论与自动机理论之间的深层联系。
On prime endomorphisms of the free group of rank two
研究秩为2的自由群F2的素自同态，给出了非素自同态对数密度的下界3/4和上界27/28，等价于随机自同态为非素的概率指数衰减的上下界。
Functional degrees and arithmetic applications III: Beyond Prime Exponent
本文将素数 Ax-Katz 定理推广到任意有限交换群之间的多个映射，用不变因子和函数次数给出联立零点集基数的 p-adic 可除性下界。
This note improves the best known exponent 1/12 in the prime field sum-product inequality (for small sets) to 1/11, modulo a logarithmic factor.
We study the commuting graph on elements of odd prime order in finite simple groups. The results are used in a forthcoming paper describing the structure of Bruck loops and Bol loops of exponent 2.
This paper studies the MOR cryptosystem, using the automorphism group of the extra-special $p$-group of exponent $p$, for an odd prime $p$. Similar results can be obtained for extra-special $p$-groups of exponent $p^2$ and for the even prime.
Sums of binomial coefficients modulo $p$ and groups of exponent $p^n$
本文利用矩阵方法证明模素数 $p$ 的二项式系数和同余方程，并据此构造指数为 $p^n$ 的群，从而说明 $p=2$ 在 Heisenberg 群相关问题上并非真正例外。
On the generation of multiplicative groups by small primes
该研究受Regev量子因式分解算法启发，证明存在至多$(\log Q)^{1+\varepsilon}$个不超过$(\log Q)^{C_*(A+1)}$的小素数，使得几乎所有$q\leq Q$的模$q$乘法群元素可由这些素数的子集生成，且素数个数指数最优。
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